mathematics, specifically in category theory, a functor F : C → D {\displaystyle F:C\to D} is essentially surjective if each object d {\displaystyle d} of D {\displaystyle...

morphisms between them. Thus any functor from C {\displaystyle C} to E {\displaystyle E} will not be essentially surjective. Consider a category C {\displaystyle...

{G}}(U)} are not always surjective for epimorphisms of sheaves is equivalent to non-exactness of the global sections functor—or equivalently, to non-triviality...

anafunctor. For example, the statement "every fully faithful and essentially surjective functor is an equivalence of categories" is equivalent to the axiom...

are functions preserving this structure. There is a natural forgetful functor U : Top → Set to the category of sets which assigns to each topological...

bifunctor; but as (single) functor, of type [ X , − ] {\displaystyle [X,-]} , it appears as an adjoint functor to a functor of type − × X {\displaystyle...

which is an abelian category equipped with an exact functor from A to A/B that is essentially surjective and has kernel B. This quotient category can be constructed...

1 ( y ) = { x } . {\displaystyle f^{-1}(y)=\{x\}.} The function f is surjective (or onto, or is a surjection) if its range f ( X ) {\displaystyle f(X)}...

is unique up to equivalence. First, this functor F ~ {\displaystyle {\tilde {F}}} is essentially surjective if any object in D can be expressed as a filtered...

if there is an equivalence between them. essentially surjective A functor F is called essentially surjective (or isomorphism-dense) if for every object...

category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category...

(injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections...

are the surjective measurable maps, and the isomorphisms are the isomorphisms of measurable spaces. The split monomorphisms are (essentially) the inclusions...

{B}}\cong {\mathcal {C}}} if and only if there exists an exact and essentially surjective functor F : A → C {\displaystyle F\colon {\mathcal {A}}\to {\mathcal...

groups 1 → A → H → G → 1 {\displaystyle 1\to A\to H\to G\to 1\!} the surjective homomorphism d : H → G {\displaystyle d\colon H\to G\!} together with...

finitely generated, projective C∞(M)-modules is full, faithful, and essentially surjective. Therefore the category of smooth vector bundles on M is equivalent...

(this definition is motivated by a descent theorem of Grothendieck for surjective étale maps of affine schemes). With these definitions, the algebraic spaces...

pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar...

equivalence for each pair of objects x , y {\displaystyle x,y} , and essentially surjective, meaning for each object y in D, y ≃ F ( x ) {\displaystyle y\simeq...

is extra structure. Essentially, these are groupoids G 1 , G 0 {\displaystyle {\mathcal {G}}_{1},{\mathcal {G}}_{0}} with functors s , t : G 1 → G 0 {\displaystyle...

sheaves, and a surjective map corresponds to an injection of sheaves. An alternative approach to the dual point of view is to use the functor of points. If...

of order 2 as above, the Hilbert functor HilbV/C of closed subschemes is not representable by a scheme, essentially because the quotient by the group...

formal language, is a natural-valued function. Computability theory is essentially based on natural numbers and natural (or integer) functions on them....

pair of objects a , b {\displaystyle a,b} . f {\displaystyle f} is essentially surjective if for each object y {\displaystyle y} in Y {\displaystyle Y} ,...

to C) is all of C; that is, if and only if that map is an epimorphism (surjective, or onto). Therefore, the sequence 0 → X → Y → 0 is exact if and only...

{m}}:M/{\mathfrak {m}}M\to N/{\mathfrak {m}}N} is surjective, then ϕ {\displaystyle \phi } is surjective. Nakayama's lemma also has several versions in homological...

S. This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids. Thus, an...

f {\displaystyle G/\ker f} . In particular, if f {\displaystyle f} is surjective then H {\displaystyle H} is isomorphic to G / ker ⁡ f {\displaystyle G/\ker...

that monotone Galois connections are special cases of pairs of adjoint functors in category theory as discussed further below. Other terminology encountered...

or surjective, then the same is true for all above defined linear maps. In particular, the tensor product with a vector space is an exact functor; this...